Point-to multipoint (P2MP) network resource management

ABSTRACT

Techniques for managing resources in a point-to-multipoint (P2MP) network are disclosed. In some examples, a root station is adapted to transmit and receive network packets and leaf stations are adapted to transmit and receive the network packets from the root station. An electrical control system can be adapted to adjust a control error toward a zero value and adjust an output toward a steady state. The electrical control system may include feedback to control the root station based, at least in part, on the output of the electrical control system.

BACKGROUND

Point to multipoint (P2MP) topology is one of the most commonly usedtopologies in an access network. In general, P2MP may include a rootstation (RS) and a number of leaf stations (LSs). In P2MP, any mediahaving a RS that broadcasts packets through a single trunk (such as afrequency, wavelength, or wireless channel) to LSs typically may bereferred to as downstream. Similarly, LSs unicasting packets throughbranches and the trunk to the RS may be referred to as upstream. Inaddition, the LSs may not communicate with each other in a peer-to-peermanner.

Many wired broadband access networks such as the Time Division Multiplex(TDM) Passive Optical Network (PON) (which includes Ethernet passiveoptical networks (EPONs), Gigabit passive optical networks (GPONs), andBroadband passive optical networks (BPONs)), can be generalized into aP2MP architecture. The P2MP architecture of PONs may reduce the dominantdeployment and maintenance cost, and facilitates the central managementby utilizing the RS as the central office.

In the recent past, there have been attempts to address upstreamresource management and allocation mechanism issues in P2MP networks,especially in P2MP EPON networks. These schemes may be categorized intothree categories: fixed resource allocation (FRA), request-basedresource allocation (RRA), and prediction-based resource allocation(PRA). Although most of the schemes address the resource management inEPONs, they can be generalized to other P2MP networks by employingappropriate MAC control cells and fields in the frames.

Although attempts have been made to address the upstream resourcemanagement issue in P2MP networks, few attempts have addressed the abovedifferent resource management schemes such that these schemes can beevaluated, compared and further improved. Furthermore, few currentupstream resource allocation schemes in a P2MP system address queuemanagement, which may have a significant impact in achieving highnetwork resource utilization.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

The foregoing and other features of the present disclosure will becomemore fully apparent from the following description and appended claims,taken in conjunction with the accompanying drawings. Understanding thatthese drawings depict only several embodiments in accordance with thedisclosure and are, therefore, not to be considered limiting of itsscope, the disclosure will be described with additional specificity anddetail through use of the accompanying drawings.

FIG. 1 is a diagram of a general P2MP architecture;

FIG. 2 is a flowchart showing the operation of an example embodiment;

FIG. 3 is a flowchart showing the operation of another exampleembodiment;

FIG. 4 is a flowchart showing the operation of yet another exampleembodiment;

FIG. 5 is a diagram depicting service cycles over time in an exampleembodiment of a P2MP architecture;

FIG. 6 is a schematic diagram of an example embodiment of a controlsystem;

FIG. 7 is a schematic diagram of another example embodiment of a controlsystem; and

FIG. 8 is a block diagram of an example computing device that isarranged for point-to-multipoint (P2MP) network resource management, allarranged in accordance with the present disclosure.

DETAILED DESCRIPTION

In the following detailed description, reference is made to theaccompanying drawings, which form a part hereof. In the drawings,similar symbols typically identify similar components, unless contextdictates otherwise. The illustrative embodiments described in thedetailed description, drawings, and claims are not meant to be limiting.Other embodiments may be utilized, and other changes may be made,without departing from the spirit or scope of the subject matterpresented here. It will be readily understood that the aspects of thepresent disclosure, as generally described herein, and illustrated inthe Figures, can be arranged, substituted, combined, and designed in awide variety of different configurations, all of which are explicitlycontemplated and make part of this disclosure.

This disclosure is drawn, inter alia, to methods and systems related tosystems and methods for efficient resource management in P2MP networks.

FIG. 1 is a diagram of a general point-to-multipoint (P2MP) architecturearranged in accordance with the present disclosure. As show in FIG. 1, acontrol system for use with a point-to-multipoint (P2MP) networkresource management system may include at least one root station 10(identified in FIG. 1 as RS-1, RS-2 and RS-n, where n may be anynumeral) adapted to transmit and receive network packets, and at leastone leaf station 12 (identified in FIG. 1 as LS-1, LS-2 and LS-n, wheren may be any numeral) may be adapted to transmit and receive networkpackets from the at least one root station 10. Each leaf station 12 maybe in communication with the root station(s) 10. The control system mayinclude a resource management component 14 adapted to manage networkpacket transmission that may include a reference input signal 16, acompensator 18, a comparator 20, a controller 22, and a feedback loop24.

The compensator 18 may be operably connected to the reference inputsignal 16. The compensator 18 may be adapted to offset a control error(which may be the difference between the reference input signal 16 andan output signal 26, for example). The compensator 18 may be furtheradapted to drive the control error toward a zero value. The comparator20 may be adapted to calculate the control error and may be operablyconnected to the compensator 18. The controller 22 may be adapted tooutput the output signal 26. Further, the controller 22 may be adaptedto manipulate the output signal 26 if the control error is determined tobe a non-zero value. The feedback loop 24 may be operably connected tothe controller 22 and the comparator 20. Further, the feedback loop 24may include a controller gain (which may be multiplied by or otherwisecombined with the output signal 26).

In some example embodiments, the controller 22 may be configured toadjust the controller gain to manipulate the output signal 26 such thatspecific characteristics of the output signal 26 may be attained. Designobjectives for resource allocation in a P2MP system may include systemrobustness, accuracy, and target transient performance. These objectives(and others) may be achieved through proper design of the compensator20, as discussed below.

Some additional example embodiments may include a method for anelectrical control system to manage resources in a point-to-multipoint(P2MP) network, which may operate as depicted by the flowchart in FIG.2. The illustrated embodiment may include one or more of processingoperations 28, 30 and 32. Operation 28 may include controlling the atleast one root station based, at least in part, on output of theelectrical control system, wherein the electrical control system isconfigured to reduce an amount of time for the electrical control systemto produce an output comprising a steady state and also configured todefine a maximum boundary for the output, the electrical control systemcomprising a feedback control loop and a compensator, wherein thecompensator is adapted to adjust a control error toward a zero value andto adjust the output toward the steady state. Operation 30 may includetransmitting the one or more network packets to the at least one leafstation, wherein the at least one root station is capable oftransmitting and receiving the one or more network packets from the atleast one root station, and wherein the at least one leaf station iscapable of communication with the at least one root station. Operation32 may include repeating the controlling and transmitting for at leastone of the one or more network packets.

In some example embodiments, a point-to-multipoint (P2MP) networkarchitecture may be configured to implement the method of FIG. 2.

Additional example embodiments may include a method for an electricalcontrol system to manage resources in a point-to-multipoint (P2MP)network, which may operate as depicted by the flowchart in FIG. 3. Theexample embodiment may include one or more of processing operations 34,36, 38, 40 and 42. Operation 34 may include adapting a compensator forthe electrical control system based, at least in part, on the equation

${F_{i} = {{\lbrack {K_{i}\mspace{25mu} 1} \rbrack\begin{bmatrix}{A_{i} - I} & B_{i} \\C_{i} & 0\end{bmatrix}}^{- 1}\begin{bmatrix}0 \\1\end{bmatrix}}},{{where}\begin{bmatrix}{A_{i} - I} & B_{i} \\C_{i} & 0\end{bmatrix}}$is a non-singular matrix, where A_(i) is a state matrix, where B_(i) isan input matrix, where C_(i) is an output matrix, and where K_(i) is acontroller gain for adjusting the output, wherein the electrical controlsystem includes a controller gain for an output, a compensator foradjusting a control error toward a zero value and adjusting the outputtoward a steady state. Operation 36 may include repeatedly analyzing theoutput of the electrical control system to determine if the output isequal to a desired output. Operation 38 may include dynamicallyadjusting the output of the electrical control system to provide a newoutput by changing the controller gain until the new output achieves thedesired output. Operation 40 may include transmitting one or morenetwork packets to one or more of the at least one leaf station and atleast one root station based at least in part on the output of theelectrical control system. At operation 42, the controller gain isK_(i), a constant matrix and may be defined by the equation

$\{ \begin{matrix}{{f_{1}( {k_{11},k_{12},k_{21},k_{22},\alpha_{i}} )} = {{- 2}{\mathbb{e}}^{{- 4}/T_{i}}{\cos( {\pi\frac{\log\; r}{\log\; M_{i}}} )}}} \\{{{f_{2}( {k_{11},k_{12},k_{21},k_{22},\alpha_{i}} )} = {\mathbb{e}}^{{- 2}/T_{i}}},}\end{matrix} $where k₁₁, k₁₂, k₂₁ and k₂₂ are vectors of K_(i), α_(i) is an estimateindex, r is a reference input, T_(i) is the amount of time for theelectrical control system to produce an output having a steady state,and M_(i) is the maximum boundary for the output.

In some example embodiments, a point-to-multipoint (P2MP) networkarchitecture may be configured to implement the method of FIG. 3. In oneexample, such a point-to-multipoint network architecture may implementthe PRA scheme.

Additional example embodiments may include a method for an electricalcontrol system to manage resources in a point-to-multipoint (P2MP)network, which may operate as depicted by the flowchart in FIG. 4. Theexample embodiment may include one or more of processing operations 44,46, 48, 50 and 52. Operation 44 may include adapting a compensator forthe electrical control system based, at least in part, on the equation

${F_{i} = {{\lbrack {K_{i}\mspace{25mu} 1} \rbrack\begin{bmatrix}{A_{i} - I} & B_{i} \\C_{i} & 0\end{bmatrix}}^{- 1}\begin{bmatrix}0 \\1\end{bmatrix}}},{{where}\begin{bmatrix}{A_{i} - I} & B_{i} \\C_{i} & 0\end{bmatrix}}$is a non-singular matrix, where A_(i) is a state matrix, where B_(i) isan input matrix, where C_(i) is an output matrix, and where K_(i) is acontroller gain for adjusting the output, wherein the electrical controlsystem includes a controller gain for an output, a compensator foradjusting a control error toward a zero value and adjusting the outputtoward a steady state. Operation 46 may include repeatedly analyzing theoutput of the electrical control system to determine if the output isequal to a desired output. Operation 48 may include dynamicallyadjusting the output of the electrical control system to provide a newoutput by changing the controller gain until the new output achieves thedesired output. Operation 50 may include transmitting one or morenetwork packets to one or more of the at least one leaf station and atleast one root station based at least in part on the output of theelectrical control system. At operation 52, the controller gain isK_(i), a constant matrix and may be defined by the equation

$\{ \begin{matrix}{{f_{1}( {k_{1},k_{2}} )} = {{- 2}{\mathbb{e}}^{{- 4}/T_{i}}{\cos( {\pi\frac{\log\; r}{\log\; M_{i}}} )}}} \\{{{f_{2}( {k_{1},k_{2}} )} = {\mathbb{e}}^{{- 2}/T_{i}}},}\end{matrix} $where k₁ and k₂ are vectors of K_(i), r is a reference input, T_(i) isthe amount of time for the electrical control system to produce anoutput comprising a steady state, and M_(i) is the maximum boundary forthe output.

In some example embodiments, a point-to-multipoint (P2MP) networkarchitecture may be configured to implement the method of FIG. 4. In oneexample, such a point-to-multipoint network architecture may implementthe RRA scheme.

The present disclosure now considers a P2MP system with one RS and yLSs, as depicted by the diagram in FIG. 5. The RS serves each LS once ina service cycle 66, 68, 70. As previously discussed, the presentdisclosure contemplates that an issue with P2MP networks may includequeue management for upstream resource management. A purpose of queuemanagement may be to ensure a desired queue length Q_(i) ^(d) in theLSs. The desired queue length Q_(i) ^(d) may be defined as the efficientqueue length to maintain high network resource utilization.Theoretically, each LS may need to maintain a desired queue length Q_(i)^(d) to avoid overflow or emptiness. If the queue length is too large,data loss and retransmission may occur due to a limited amount ofavailable buffers. On the other hand, if the queued length becomesempty, it may indicate that the allocated resource for this LS may bemore than it actually needs. The network resource may thus be wastedwith low utilization. It may be desirable to avoid both extremes bymaintaining a desired queue length Q_(i) ^(d).

The upstream resource could be bandwidth (TDM-PON), wavelength(wavelength-division multiplexing or WDM-PON), or frequency (orthogonalfrequency-division multiplexing or OFDM). Under the P2MP architecture,multiple LSs may share an upstream trunk, and each LS may have noknowledge of the transmission condition of the other LSs. To avoid datacollision, a request/grant arbitrary mechanism, such as the multipointcontrol protocol (MPCP) in an EPON, may be deployed for upstreamresource sharing. The request/grant mechanism may be implemented incontinuous service cycles 66, 68, 70. In each service cycle 66, 68, 70,LSs send requests 72 to RS for a resource grant 74 before anytransmission may occur. Thereafter, RS may determine an appropriatetransmission window in the next service cycle 68, 70 to each LS, byconsidering the requests 72 as well as the available resources, and maysend out grants 74 to LSs. Finally, after receiving the grants 74, LSsmay begin to transmit their packets until their granted window haspassed. In this manner, a dynamic resource allocation may be achieved.

As used herein, the following notations shall be adopted.

-   Q_(i)(n): the reported residual queued length from LS_(i)(1≦l≦y) at    the end of service cycle n 66;-   R_(i)(n): the resource request 72 of LS_(i) for service cycle n 66    (R_(i)(n) may or may not be the same as Q_(i)(n), depending on the    particular resource allocation scheme as described below);-   λ_(i)(n): the actual arrived data of LS_(i) at service cycle n+1 68;-   {circumflex over (λ)}_(i)(n): the predicted arrival data at LS_(i)    in service cycle n+1 68;-   d_(i)(n): the departed data from LS_(i) at service cycle n 66;-   G_(i)(n): the allocated timeslot to LS_(i) at service cycle n 66;-   G_(i) ^(max): the maximum timeslot length prescribed by the service    level agreement (SLA).

Since no queue status report may be conducted in the FRA scheme, and thereported queue length may equal zero, e.g.,Q _(i)(n+1)=0  (Eq. 1a)for FRA.

In some RRA schemes, the reported queue length of transmission cycle(n+1) 68 may be determined by the difference of the injected data, whichmay include the transmission residual of cycle n 66 (e.g., Q_(i)(n)) aswell as the incoming data arrived in the waiting time at ONU_(i) intransmission cycle n 66 (e.g., λ_(i)(n)), and the delivered data (e.g.,d_(i)(n)), e.g.,Q _(i)(n+1)=Q _(i)(n)+λ_(i)(n)−d _(i)(n+1)  (Eq. 1b)

In some PRA schemes, “over-grant” may occur. This “over-grant” may beadjusted by reporting the difference between the injected data (e.g.,Q_(i)(n)+λ_(i)(n)) and the grant G_(i)(n) 74, e.g.,Q _(i)(n+1)=Q _(i)(n)+λ_(i)(n)−G _(i)(n+1)  (Eq. 1c)Eqs. 1a-1c may be summarized as

$\begin{matrix}{{Q_{i}( {n + 1} )} = \{ \begin{matrix}{0,} & {{for}\mspace{14mu}{FRA}} \\{{{Q_{i}(n)} + {\lambda_{i}(n)} - {d_{i}( {n + 1} )}},} & {{for}\mspace{14mu}{RRA}} \\{{{Q_{i}(n)} + {\lambda_{i}(n)} - {G_{i}( {n + 1} )}},} & {{for}\mspace{14mu}{PRA}}\end{matrix} } & ( {{Eq}.\mspace{14mu} 1} )\end{matrix}$

On the other hand, the resource request R_(i)(n) 72 of LS_(i) forservice cycle n 66 may be determined by perspective resource allocationschemes. For FRA, the resource request of LSi in service cycle (n+1) 68(e.g., R_(i)(n+1)) is the fixed value R_(fix), e.g.,R _(i)(n+1)=R _(fix)  (Eq. 2a)In RRA, R_(i)(n+1) may be determined by the reported queue length, e.g.,R _(i)(n+1)=Q _(i)(n),  (Eq. 2b)When a traffic predictor is employed, as in PRA, R_(i)(n+1) may bedetermined by the sum of the reported queue length and the predictedarrival data, e.g.,R _(i)(n+1)=Q _(i)(n)+{circumflex over (λ)}_(i)(n)  (Eq. 2c)where {circumflex over (λ)}_(i)(n) is the predicted arrival data atLS_(i) in service cycle (n+1) 68. Eq. 2a-2c may be summarized as

$\begin{matrix}{{R_{i}( {n + 1} )} = \{ \begin{matrix}{R_{fix},} & {{for}\mspace{14mu}{FRA}} \\{{Q_{i}(n)},} & {{for}\mspace{14mu}{RRA}} \\{{{Q_{i}(n)} + {{\hat{\lambda}}_{i}(n)}},} & {{for}\mspace{14mu}{PRA}}\end{matrix} } & ( {{Eq}.\mspace{14mu} 2} )\end{matrix}$

After processing the request 72, the RS allocates time windowsG_(i)(n+1) to LS_(i). In FRA, the assigned resource to LSi intransmission cycle n+1 68 (e.g., G_(i)(n+1)) may be the fixed valueR_(fix). In both RRA and PRA, G_(i)(n+1) may be the smaller value of thebandwidth request (e.g., R_(i)(n+1)) and the SLA parameter (e.g., G_(i)^(max)), e.g.,

$\begin{matrix}{{G_{i}( {n + 1} )} = \{ \begin{matrix}{R_{fix},} & {{for}\mspace{14mu} F\; R\; A} \\{{\min\lbrack {{R_{i}( {n + 1} )},G_{i}^{\max}} \rbrack},} & {{for}\mspace{14mu} R\; R\; A} \\{{\min\lbrack {{R_{i}( {n + 1} )},G_{i}^{\max}} \rbrack},} & {{for}\mspace{14mu} P\; R\;{A.}}\end{matrix} } & ( {{Eq}.\mspace{14mu} 3} )\end{matrix}$

After receiving the bandwidth allocation decision, LS_(i) may scheduleits upstream transmission indicated by G_(i)(n+1), and the delivereddata d_(i)(n+1) may be described asd _(i)(n+1)=min{G _(i)(n+1),Q _(i)(n)+λ_(i)(n)}  (Eq. 4).

The present disclosure considers that a unified state space model may beconstructed for FRA, RRA, and PRA based, at least in part, on Eqs. 1 and2, as followsX _(i)(n+1)=AX _(i)(n)+BU _(i)(n),  (Eq. 5)where X_(i)(n)=[R_(i)(n) Q_(i)(n)]^(T) may be the state vector,indicating the bandwidth requirement and the queue length of LS_(i), andU_(i)(n) may be the input vector, representing the arrived data duringthe waiting time and the SLA parameter. A and B may be the matrices forthe state vector and input vector, respectively, that may determineintrinsic characteristics of each scheme at the system level.

Therefore, a unified model for upstream resource allocation over a P2MPsystem may be established through the state space equation (Eq. 5), withEqs. 3 and 4 being performance constraints. The model essentiallyexhibits the relationship between the input (e.g., on-line networktraffic load), output (e.g., bandwidth allocation decision), and statevariables (e.g., queue length and resource requirement). The state spacerepresentation may provide a convenient and compact way to model andanalyze various resource allocation schemes for the P2MP system from thecontrol theory point of view. In this way, a specific resourceallocation scheme may essentially define its particular coefficientmatrices A and B to assign the upstream resource in a different way.

The present disclosure contemplates that the above control objectivesmay have been difficult to solve in P2MP systems because of thecomplexity of mapping the objectives into the corresponding schedulingalgorithm and resource management schemes. However, the example statespace model described herein may give a simple and straightforwardframework to achieve such objectives by using state space feedbackcontrol techniques.

In some example embodiments, the settling time T_(i) and the maximumovershoot M_(i) may be two central parameters for the transientperformance. The settling time T_(i) may be defined as the time for theP2MP system to reach the steady state. Short settling times may beutilized to achieve the performance objective, especially when theincoming traffics of LSs have large volatility. In such case, shortsettling time may ensure that the system converges to the stable statebefore the traffic load changes. On the other side, the maximumovershoot M_(i) may be defined as the difference between the maximumsystem output y_(max) and steady-state system output y_(ss) divided bythe steady-state system output y_(ss), e.g.,

$\begin{matrix}{M_{i} = \frac{y_{\max} - y_{ss}}{y_{ss}}} & ( {{Eq}.\mspace{14mu} 6} )\end{matrix}$The maximum overshoot may give the upper bound for the outputoscillations of a P2MP system. For example, the specifications of a P2MPsystem may call for the system to reach a stable state within 10seconds, and the overshoot should be less than 5%.

In some embodiments, for resource allocation schemes that may be based,at least in part, on the state space model (e.g., Eq. (5)), there mayexist a controller, u_(i)(n) 76, such thatu _(i)(n)=−K _(i) x _(i)(n),  (Eq. 7)to drive the system into a close-loop form, as long as the system iscontrollable; this is known as pole placement. Substituting Eq. 7 intoEq. 5 yields,x _(i)(n+1)=(A _(i) −B _(i) K _(i))x _(i)(n).  (Eq. 8)which is the close-loop form for Eq. 5, where K_(i) 78 may be a constantmatrix. The controller design in this embodiment is illustrated in FIG.6.

The present disclosure contemplates that, from the control point ofview, the settling time and maximum overshoot may be determined by theclosed loop poles. Further, the controller gain K_(i) 78 in Eq. 8 mayessentially determine the poles in the closed loop characteristicpolynomial det[zI−(A_(i)−B_(i)K_(i))]. Thus, the target transientperformance T_(i) and M_(i) may be achieved by properly tuning thecontroller gain K_(i) 78.

Now, the present disclosure considers that the poles of a second orderP2MP system are a pair of complex conjugates re^(±jθ). According tocontrol theory, the relationship between the pole parameters r and θ,and the settling time T_(i) and the maximum overshoot M_(i) may bestated as

$\begin{matrix}{r \approx {\mathbb{e}}^{{- 4}/T_{i}}} & ( {{{Eq}.\mspace{14mu} 9}a} ) \\{\theta \approx {\pi\frac{\log\; r}{\log\; M_{i}}}} & ( {{{Eq}.\mspace{14mu} 9}b} )\end{matrix}$The eigenvalues of the closed-form characteristic polynomialdet[zI−(A_(i)−B_(i)K_(i))] may be re^(±jθ), or simply,det[zI−(A_(i)−B_(i)K_(i))]=(z−re^(jθ))(z−re^(−jθ)), e.g.,det[zI−(A _(i) −B _(i) K _(i))]=z ²−2r cos θz+r ²  (Eq. 10)

On the other hand, the present disclosure considers that the secondorder closed-form characteristic polynomial for the PRA scheme may bebased, at least in part, on,det[zI−(A _(i) −B _(i) K _(i))]=z ² +f ₁(k ₁₁ ,k ₁₂ ,k ₂₁ ,k ₂₂,α_(i))z+f ₂(k ₁₁ ,k ₁₂ ,k ₂₁ ,k ₂₂,α_(i))  (Eq. 11)where k₁₁, k₁₂, k₂₁ and k₂₂ are vectors of K_(i), and α_(i) is theestimate index.

As Eqs. 10 and 11 represent the same closed-form characteristicpolynomial for PRA, they have the same coefficients for each order of z.Thus,

$\begin{matrix}\{ \begin{matrix}{{f_{1}( {k_{11},k_{12},k_{21},k_{22},\alpha_{i}} )} = {{- 2}r\;\cos\;\theta}} \\{{f_{2}( {k_{11},k_{12},k_{21},k_{22},\alpha_{i}} )} = {r^{2}.}}\end{matrix}  & ( {{Eq}.\mspace{14mu} 12} )\end{matrix}$By substituting Eqs. 9a and 9b into Eq. 12, the result is

$\begin{matrix}\{ \begin{matrix}{{f_{1}( {k_{11},k_{12},k_{21},k_{22},\alpha_{i}} )} = {{- 2}{\mathbb{e}}^{{- 4}/T_{i}}{\cos( {\pi\frac{\log\; r}{\log\; M_{i}}} )}}} \\{{f_{2}( {k_{11},k_{12},k_{21},k_{22},\alpha_{i}} )} = {\mathbb{e}}^{{- 2}/T_{i}}}\end{matrix}  & ( {{Eq}.\mspace{14mu} 13} )\end{matrix}$

Eq. 13 provides the range of each vector of the controller gain K_(i) 78to reach the target settling time and maximum overshoot. The solutionsof Eq. 13 also show relationships between each vector of the controlgain matrix 78 and the estimate index. Although the exact value of eachvector and estimate index is not given, Eq. 13 may essentially provide aguideline to design a suitable controller gain K_(i) 78 such that thetarget settling time T_(i) and maximum overshoot M_(i) in the PRA schememay be met. It is also noted that the estimate index α_(i) may have animpact on achieving the called for transient performance.

Similarly, the characteristic polynomial for the RRA scheme may bedet[zI−(A _(i) −B _(i) K _(i))]=z ² +f ₁(k ₁ ,k ₂)z+f ₂(k ₁ ,k ₂)  (Eq.14)where k₁ and k₂ are vectors of K_(i) 78.

Since Eqs. 10 and 14 represent the same or similar closed-formcharacteristic polynomial for RRA, they may have the same or similarcoefficients for each order of z. Comparing the coefficients of Eqs. 10and 14 yields

$\begin{matrix}\{ \begin{matrix}{{f_{1}( {k_{1},k_{2}} )} = {{- 2}r\;\cos\;\theta}} \\{{f_{2}( {k_{1},k_{2}} )} = r^{2}}\end{matrix}  & ( {{Eq}.\mspace{14mu} 15} )\end{matrix}$

Substituting Eqs. 9a and 9b into Eq. 15 yields

$\begin{matrix}\{ \begin{matrix}{{f_{1}( {k_{1},k_{2}} )} = {{- 2}{\mathbb{e}}^{{- 4}/T_{i}}{\cos( {\pi\frac{\log\; r}{\log\; M_{i}}} )}}} \\{{f_{2}( {k_{1},k_{2}} )} = {\mathbb{e}}^{{- 2}/T_{i}}}\end{matrix}  & ( {{Eq}.\mspace{14mu} 16} )\end{matrix}$Therefore, the solutions of Eq. 16 may essentially provide guidelines todesign a suitable controller gain K_(i) 78 such that the target settlingtime T_(i) and maximum overshoot M_(i) in RRA scheme may be met.

FIG. 6 is a schematic diagram of an example embodiment of a controlsystem arranged in accordance with the present disclosure. As shown inFIG. 6, the target system 80 may be achieved by feeding backproportional state variables 84 to the control input 82. The statevariables 84 may represent the on-line traffic dynamics, which may implychanges of the queue length and bandwidth requirement of an LS. Thecontroller 76 may essentially feedback the traffic dynamics information,after multiplying by the controller gain 78, to the input 82 of thesystem. By doing so, the eigenvalues of an open plant system, which isusually outside of the unit circle, may be driven back into the insideof the unit circle after implementing proper controller gains. In thismanner, the system is driven into the stable state. An examplecontroller 76 may be facilitated through the proper buffering andintra-LS scheduling schemes at the RS, or the appropriate inter-LSscheduling scheme among LSs. Thus, the RS may work as a centralcontroller to tune LSs accordingly, which may ensure that the upstreamresource of a P2MP system is fairly shared among multiple LSs. Thecontroller gains K_(i)|_(i=1,2,3,4,5,6) 78 describe the controller 76characteristics in different scenarios.

For PRA, the RS may manipulate the upstream transmission from multipleLSs by using a controller U_(i)(n)=−KX_(i)(n) 76, where K=K_(i)|_(i=1˜4)78. The estimation index α_(i) may affect the system stability whendesigning a controller 76 for a P2MP system with PRA. In both RRA andPRA, the above equations may provide guidelines for the controller 76design to achieve a P2MP system's stability.

Based, at least in part, on the models discussed above, example controlsystem designs may now be determined. An example of one such controlsystem design is depicted in FIG. 7. The present disclosure considersthat design objectives typically may be an initial step in controlsystem design. In this case, the design objectives for resourceallocation in a P2MP system may be system robustness, accuracy, andtarget transient performance.

In designing a control system in one embodiment, consider the measuredsystem output Y_(i)(n)=CX_(i)(n) 86, and define matrix c=[0 1]. Thesystem output essentially may be the measurement to the report queuelength Q_(i)(n). The state space system may then be described asX _(i)(n+1)=AX _(i)(n)+BU _(i)(n)₉₀  (Eq. 17)Y _(i)(n)=CX _(i)(n)It may be useful to design a compensator 88 to achieve the designobjectives of desired queue length Q_(i) ^(d). FIG. 7 is a schematicdiagram illustrating another example embodiment of a control systemarranged in accordance with the present disclosure. As illustrated, anexample compensator 88 may be realized by implementing the controllerU _(i)(n)=−K _(i) X _(i)(n)+F _(i) ^(r)92  (Eq. 18)to achieve the prescribed objectives. The reference input r 82 is thedesired queue length, and thus e(n)=Y_(i)(n)−r is the control error. Thematrix F_(i) 88 is a compensator to offset the control error, so thatthe system output 86 can eventually converge to the input reference 82(e.g., e(n)=0). Therefore, a suitable compensator F_(i) 88 may bedetermined.

As discussed above, the objective of queue management for achieving thedesired queue length Q_(i) ^(d) 82 may be mapped to how to design thecompensator F_(i) 88, such that the queue length measurement Y_(i)(n) 86may converge to the system input r 82, which may be the desired queuelength. To reach this objective, in some embodiments, a compensatorF_(i) 88 may be implemented immediately after the reference 82, andadded up to the feedback from the state variable 84, to form thecontroller U_(i)(n) 92, which is illustrated in FIG. 7. Therefore, acompensator F_(i) 88 may be designed in such way to offset the controlerror, e.g., e(n)=0.

For a particular P2MP system i, the compensator F_(i) 88 may bedetermined by the state matrix A_(i), the input matrix B_(i), the outputmatrix C_(i), and the controller gain K_(i) 94. The present disclosurecontemplates that the compensator F_(i) 88 that drives the control errore(n)=Y_(i)(n)−r to zero may be given by

$\begin{matrix}{{F_{i} = {{\lbrack {K_{i}\mspace{20mu} 1} \rbrack\begin{bmatrix}{A_{i} - I} & B_{i} \\C_{i} & 0\end{bmatrix}}^{- 1}\begin{bmatrix}0 \\1\end{bmatrix}}}{{where}\begin{bmatrix}{A_{i} - I} & B_{i} \\C_{i} & 0\end{bmatrix}}} & ( {{Eq}.\mspace{14mu} 19} )\end{matrix}$is a non-singular matrix.

Eq. 19 may be derived by the following process. When a system outputconverges to the input reference 82 (e.g., e(n)=Y_(i)(n)−r=0), the statevariable X_(i)(n) 84 may reach its steady state X_(i) ^(ss). Assume theassociated steady state input is U_(i) ^(ss) the controller representedby Eq. 18 may be re-written asU _(i)(n)=−K _(i)(X _(i)(n)−X _(i) ^(ss))+U _(i) ^(ss)  (Eq. 20)From Eq. 20, it is shown that the system input U_(i)(n) 92 may reach itssteady state U_(i) ^(ss) when the state variables X_(i)(n) 84 reachX_(i) ^(ss). Eq. 20 may be further re-written asU_(i)(n)=−K_(i)X_(i)(n)+K_(i)X_(i) ^(ss)+U_(i) ^(ss), e.g.,

$\begin{matrix}{{U_{i}(n)} = {{{- K_{i}}{X_{i}(n)}} + {\lbrack {K_{i}\mspace{20mu} 1} \rbrack\begin{bmatrix}X_{i}^{ss} \\U_{i}^{ss}\end{bmatrix}}}} & ( {{Eq}.\mspace{14mu} 20} )\end{matrix}$

When the system reaches the steady state, the following equations mayhold true,X _(i) ^(ss) =AX _(i) ^(ss) +BU _(i) ^(ss)Y_(i) ^(ss)=CX_(i) ^(ss)  (Eq. 21)Y_(i) ^(ss)=rEq. 21 further yields, (A−I)X_(i) ^(ss)+BU_(i) ^(ss)=0 e.g.,CX_(i) ^(ss)=r

$\begin{matrix}{{\begin{bmatrix}{A - I} & B \\C & 0\end{bmatrix}\begin{bmatrix}X_{i}^{ss} \\U_{i}^{ss}\end{bmatrix}} = \begin{bmatrix}0 \\r\end{bmatrix}} & ( {{Eq}.\mspace{14mu} 22} )\end{matrix}$Therefore, from Eq. 22, the following equation may hold true,

$\begin{matrix}{\begin{bmatrix}X_{i}^{ss} \\U_{i}^{ss}\end{bmatrix} = {\begin{bmatrix}{A - I} & B \\C & 0\end{bmatrix}^{- 1}\begin{bmatrix}0 \\r\end{bmatrix}}} & ( {{Eq}.\mspace{14mu} 23} )\end{matrix}$provided that

$\quad\begin{bmatrix}{A - I} & B \\C & 0\end{bmatrix}$is a non-singular matrix. From Eqs. 20 and 23, the controller may beexpressed as follows,

$\begin{matrix}{{U_{i}(n)} = {{{- K_{i}}{X_{i}(n)}} + {{{\begin{bmatrix}K_{i} & 1\end{bmatrix}\begin{bmatrix}{A - I} & B \\C & 0\end{bmatrix}}^{- 1}\begin{bmatrix}0 \\1\end{bmatrix}}r}}} & ( {{Eq}.\mspace{14mu} 24} )\end{matrix}$

By comparing Eqs. 18 and 24, a compensator F_(i) 88 that may offset thecontrol error is Eq. 19 discussed above and as follows

$F_{i} = {{{\begin{bmatrix}K_{i} & 1\end{bmatrix}\begin{bmatrix}{A - I} & B \\C & 0\end{bmatrix}}^{- 1}\begin{bmatrix}0 \\1\end{bmatrix}}.}$

Consequently, by implementing the compensator F_(i) 88 of Eq. 19, thecontroller 92 of Eq. 18 may be able to force the system output Y_(i)(n)86 to track the reference input r 82, implying that the queue length maybe eventually driven into the desired queue length Q_(i) ^(d). Thecompensator F_(i) 88 may be facilitated through the proper buffering andintra-LS scheduling schemes at the LS, or the appropriate inter-LSscheduling scheme among LSs.

With reference to FIG. 8, depicted is a block diagram illustrating anexample computing device 800 that is arranged for point-to-multipoint(P2MP) network resource management in accordance with the presentdisclosure. In a very basic configuration 801, computing device 800typically includes one or more processors 810 and system memory 820. Amemory bus 830 can be used for communicating between the processor 810and the system memory 820.

Depending on the desired configuration, processor 810 can be of any typeincluding but not limited to a microprocessor (μP), a microcontroller(μC), a digital signal processor (DSP), or any combination thereof.Processor 810 can include one more levels of caching, such as a levelone cache 811 and a level two cache 812, a processor core 813, andregisters 814. The processor core 813 can include an arithmetic logicunit (ALU), a floating point unit (FPU), a digital signal processingcore (DSP Core), or any combination thereof. A memory controller 815 canalso be used with the processor 810, or in some implementations thememory controller 815 can be an internal part of the processor 810.

Depending on the desired configuration, the system memory 820 can be ofany type including but not limited to volatile memory (such as RAM),non-volatile memory (such as ROM, flash memory, etc.) or any combinationthereof. System memory 820 typically includes an operating system 821,one or more applications 822, and program data 824. Application 822includes a point-to-multipoint (P2MP) network resource managementalgorithm 823 that is arranged to efficiently manage network resourcesin a point-to-multipoint network. Program Data 824 includespoint-to-multipoint (P2MP) network resource management data 825. In someembodiments, application 822 can be arranged to operate with programdata 824 on an operating system 821 to effectuate the efficientmanagement of network resources. This described basic configuration isillustrated in FIG. 8 by those components within dashed line 801.

Computing device 800 can have additional features or functionality, andadditional interfaces to facilitate communications between the basicconfiguration 801 and any required devices and interfaces. For example,a bus/interface controller 840 can be used to facilitate communicationsbetween the basic configuration 801 and one or more data storage devices850 via a storage interface bus 841. The data storage devices 850 can beremovable storage devices 851, non-removable storage devices 852, or acombination thereof. Examples of removable storage and non-removablestorage devices include magnetic disk devices such as flexible diskdrives and hard-disk drives (HDD), optical disk drives such as compactdisk (CD) drives or digital versatile disk (DVD) drives, solid statedrives (SSD), and tape drives to name a few. Example computer storagemedia can include volatile and nonvolatile, removable and non-removablemedia implemented in any method or technology for storage ofinformation, such as computer readable instructions, data structures,program modules, or other data.

System memory 820, removable storage 851 and non-removable storage 852are all examples of computer storage media. Computer storage mediaincludes, but is not limited to, RAM, ROM, EEPROM, flash memory or othermemory technology, CD-ROM, digital versatile disks (DVD) or otheroptical storage, magnetic cassettes, magnetic tape, magnetic diskstorage or other magnetic storage devices, or any other medium which canbe used to store the desired information and which can be accessed bycomputing device 800. Any such computer storage media can be part ofdevice 800.

Computing device 800 can also include an interface bus 842 forfacilitating communication from various interface devices (e.g., outputinterfaces, peripheral interfaces, and communication interfaces) to thebasic configuration 801 via the bus/interface controller 840. Exampleoutput devices 860 include a graphics processing unit 861 and an audioprocessing unit 862, which can be configured to communicate to variousexternal devices such as a display or speakers via one or more A/V ports863. Example peripheral interfaces 870 include a serial interfacecontroller 871 or a parallel interface controller 872, which can beconfigured to communicate with external devices such as input devices(e.g., keyboard, mouse, pen, voice input device, touch input device,etc.) or other peripheral devices (e.g., printer, scanner, etc.) via oneor more I/O ports 873. An example communication device 880 includes anetwork controller 881, which can be arranged to facilitatecommunications with one or more other computing devices 890 over anetwork communication via one or more communication ports 882. Thecommunication connection is one example of a communication media.Communication media may typically be embodied by computer readableinstructions, data structures, program modules, or other data in amodulated data signal, such as a carrier wave or other transportmechanism, and includes any information delivery media. A “modulateddata signal” can be a signal that has one or more of its characteristicsset or changed in such a manner as to encode information in the signal.By way of example, and not limitation, communication media can includewired media such as a wired network or direct-wired connection, andwireless media such as acoustic, radio frequency (RF), infrared (IR) andother wireless media. The term computer readable media as used hereincan include both storage media and communication media.

Computing device 800 can be implemented as a portion of a small-formfactor portable (or mobile) electronic device such as a cell phone, apersonal data assistant (PDA), a personal media player device, awireless web-watch device, a personal headset device, an applicationspecific device, or a hybrid device that include any of the abovefunctions. Computing device 800 can also be implemented as a personalcomputer including both laptop computer and non-laptop computerconfigurations.

According to one embodiment, computing device 800 is coupled to anetworking environment such that the processor 810, application 822and/or program data 824 can perform with or as a point-to-multipoint(P2MP) network resource management system in accordance with embodimentsherein.

It should also be understood that, while a stated objective of variousexample embodiments disclosed herein may be to “minimize” the settlingtime or other parameters or characteristics, it is not necessary toliterally minimize any parameters or other characteristic to fall withinthe scope of any claim unless such specific objective is expresslyclaimed. Likewise, it should be understood that it is not necessary toliterally “optimize” the settling time or other parameters orcharacteristics to fall within the scope of any claim unless suchspecific objective is expressly claimed.

The herein described subject matter sometimes illustrates differentcomponents contained within, or coupled with, different othercomponents. It is to be understood that such depicted architectures aremerely examples, and that in fact many other architectures can beimplemented which achieve the same functionality. In a conceptual sense,any arrangement of components to achieve the same functionality iseffectively “associated” such that the desired functionality isachieved. Hence, any two components herein combined to achieve aparticular functionality can be seen as “associated with” each othersuch that the desired functionality is achieved, irrespective ofarchitectures or intermedial components. Likewise, any two components soassociated can also be viewed as being “operably connected”, or“operably coupled”, to each other to achieve the desired functionality,and any two components capable of being so associated can also be viewedas being “operably couplable”, to each other to achieve the desiredfunctionality. Specific examples of operably couplable include but arenot limited to physically mateable and/or physically interactingcomponents and/or wirelessly interactable and/or wirelessly interactingcomponents and/or logically interacting and/or logically interactablecomponents.

With respect to the use of substantially any plural and/or singularterms herein, those having skill in the art can translate from theplural to the singular and/or from the singular to the plural as isappropriate to the context and/or application. The varioussingular/plural permutations may be expressly set forth herein for sakeof clarity.

It will be understood by those within the art that, in general, termsused herein, and especially in the appended claims (e.g., bodies of theappended claims) are generally intended as “open” terms (e.g., the term“including” should be interpreted as “including but not limited to,” theterm “having” should be interpreted as “having at least,” the term“includes” should be interpreted as “includes but is not limited to,”etc.). It will be further understood by those within the art that if aspecific number of an introduced claim recitation is intended, such anintent will be explicitly recited in the claim, and in the absence ofsuch recitation no such intent is present. For example, as an aid tounderstanding, the following appended claims may contain usage of theintroductory phrases “at least one” and “one or more” to introduce claimrecitations. However, the use of such phrases should not be construed toimply that the introduction of a claim recitation by the indefinitearticles “a” or “an” limits any particular claim containing suchintroduced claim recitation to inventions containing only one suchrecitation, even when the same claim includes the introductory phrases“one or more” or “at least one” and indefinite articles such as “a” or“an” (e.g., “a” and/or “an” should typically be interpreted to mean “atleast one” or “one or more”); the same holds true for the use ofdefinite articles used to introduce claim recitations. In addition, evenif a specific number of an introduced claim recitation is explicitlyrecited, those skilled in the art will recognize that such recitationshould typically be interpreted to mean at least the recited number(e.g., the bare recitation of “two recitations,” without othermodifiers, typically means at least two recitations, or two or morerecitations). Furthermore, in those instances where a conventionanalogous to “at least one of A, B, and C, etc.” is used, in generalsuch a construction is intended in the sense one having skill in the artwould understand the convention (e.g., “a system having at least one ofA, B, and C” would include but not be limited to systems that have Aalone, B alone, C alone, A and B together, A and C together, B and Ctogether, and/or A, B, and C together, etc.). In those instances where aconvention analogous to “at least one of A, B, or C, etc.” is used, ingeneral such a construction is intended in the sense one having skill inthe art would understand the convention (e.g., “a system having at leastone of A, B, or C” would include but not be limited to systems that haveA alone, B alone, C alone, A and B together, A and C together, B and Ctogether, and/or A, B, and C together, etc.). It will be furtherunderstood by those within the art that virtually any disjunctive wordand/or phrase presenting two or more alternative terms, whether in thedescription, claims, or drawings, should be understood to contemplatethe possibilities of including one of the terms, either of the terms, orboth terms. For example, the phrase “A or B” will be understood toinclude the possibilities of “A” or “B” or “A and B.”

While various aspects and embodiments have been disclosed herein, otheraspects and embodiments will be apparent to those skilled in the art.The various aspects and embodiments disclosed herein are for purposes ofillustration and are not intended to be limiting, with the true scopeand spirit being indicated by the following claims.

What is claimed is:
 1. A method for an electrical control system tomanage resources in a point-to-multipoint (P2MP) network that includesat least one root station and at least one leaf station each configuredto transmit and receive one or more network packets over the P2MPnetwork, the method for the electrical control system comprising:controlling the at least one root station based, at least in part, onoutput of the electrical control system, wherein the electrical controlsystem is configured to reduce an amount of time for the electricalcontrol system to produce an output comprising a steady state and alsoconfigured to define a maximum boundary for the output, the electricalcontrol system comprising a feedback control loop and a compensator,wherein the compensator is configured to adjust a control error toward azero value and to adjust the output toward the steady state;transmitting the one or more network packets to the at least one leafstation, wherein the at least one root station is capable oftransmitting and receiving the one or more network packets from the atleast one root station, and wherein the at least one leaf station iscapable of communication with the at least one root station; andrepeating the controlling and transmitting for at least one of the oneor more network packets; and wherein the compensator is based, at leastin part, on the equation ${F_{i} = {{\begin{bmatrix}K_{i} & 1\end{bmatrix}\begin{bmatrix}{A_{i} - I} & B_{i} \\C_{i} & 0\end{bmatrix}}^{- 1}\begin{bmatrix}0 \\1\end{bmatrix}}},{{where}\mspace{14mu}\begin{bmatrix}{A_{i} - I} & B_{i} \\C_{i} & 0\end{bmatrix}}$ is a non-singular matrix, where A_(i) is a state matrix,where B_(i) is an input matrix, where C_(i) is an output matrix, andwhere K_(i) is a controller gain.
 2. The method of claim 1, wherein theelectrical control system is based, at least in part, on the equationx_(i)(n+1)=(A_(i)−B_(i)K_(i))x_(i)(n), where x_(i)(n) is a state vectorthat indicates a bandwidth requirement and queue length of the at leastone leaf station, where A_(i) is a state vector matrix, B_(i) is aninput vector matrix, where K_(i) is a constant matrix, and where n is agiven time.
 3. The method of claim 1, wherein each of the at least oneleaf station is prohibited from communicating with at least a portion ofthe other leaf stations.
 4. The method of claim 1, wherein thecontrolling further comprises: analyzing the output to determine if theoutput is equal to a desired output; if the output is not equal to thedesired output, adjusting the output by a controller gain to produce anew output; and repeating the analyzing and the adjusting until the newoutput is equal to the desired output.
 5. The method of claim 4, whereinthe controller gain is K_(i), a constant matrix.
 6. The method of claim5, wherein K_(i) is defined by the equation $\{ {\begin{matrix}{{f_{1}( {k_{11},k_{12},k_{21},k_{22},\alpha_{i}} )} = {{- 2}\;{\mathbb{e}}^{{- 4}/T_{i}}{\cos( {\pi\frac{\log\; r}{\log\; M_{i}}} )}}} \\{{{f_{2}( {k_{11},k_{12},k_{21},k_{22},\alpha_{i}} )} = {\mathbb{e}}^{{- 2}/T_{i}}},}\end{matrix}\quad} $ where k₁₁, k₁₂, k₂₁ and k₂₂ are vectors ofK_(i), α_(i) is an estimate index, r is a reference input, T_(i) is anamount of time for the electrical control system to produce an outputcomprising a steady state, and M_(i) is a maximum boundary for theoutput.
 7. The method of claim 5, wherein K_(i) is defined by theequation $\{ {\begin{matrix}{{f_{1}( {k_{1},k_{2}} )} = {{- 2}\;{\mathbb{e}}^{{- 4}/T_{i}}{\cos( {\pi\frac{\log\; r}{\log\; M_{i}}} )}}} \\{{{f_{2}( {k_{1},k_{2}} )} = {\mathbb{e}}^{{- 2}/T_{i}}},}\end{matrix}\quad} $ where k₁ and k₂ are vectors of K_(i), r is areference input, T_(i) is an amount of time for the electrical controlsystem to produce an output comprising a steady state, and M_(i) is amaximum boundary for the output.
 8. A system for an electrical controlsystem to manage resources in a point-to-multipoint network thatincludes at least one root station and at least one leaf station thatare configured to transmit and receive network packets over the P2MPnetwork, the system for the electrical control system comprising: acompensator operably coupled to a reference input, the compensatorconfigured to offset a control error, the control error being adifference between the reference input and an output signal, thecompensator further configured to drive the control error toward a zerovalue; a comparator operably coupled to the compensator, the comparatorconfigured to calculate the control error; a controller configured tooutput the output signal and further configured to manipulate the outputsignal if the control error is determined to be a non-zero value; and acontroller gain operably coupled to the controller and the comparator,the controller gain being multiplied by the output signal; wherein thecontroller may adjust the controller gain to manipulate the outputsignal such that specific characteristics of the output signal may beattained; and wherein the compensator is based, at least in part, on theequation ${F_{i} = {{\begin{bmatrix}K_{i} & 1\end{bmatrix}\begin{bmatrix}{A_{i} - I} & B_{i} \\C_{i} & 0\end{bmatrix}}^{- 1}\begin{bmatrix}0 \\1\end{bmatrix}}},{{where}\mspace{14mu}\begin{bmatrix}{A_{i} - I} & B_{i} \\C_{i} & 0\end{bmatrix}}$ is a non-singular matrix, where A_(i) is a state matrix,where B_(i) is an input matrix, where C_(i) is an output matrix, andwhere K_(i) is a controller gain.
 9. The system of claim 8, wherein theat least one leaf station is prohibited from communicating with theother leaf stations.
 10. The system of claim 8, wherein the controllergain is K_(i), a constant matrix.
 11. The system of claim 8, wherein thecontroller gain is defined by the equation $\{ {\begin{matrix}{{f_{1}( {k_{11},k_{12},k_{21},k_{22},\alpha_{i}} )} = {{- 2}\;{\mathbb{e}}^{{- 4}/T_{i}}{\cos( {\pi\frac{\log\; r}{\log\; M_{i}}} )}}} \\{{{f_{2}( {k_{11},k_{12},k_{21},k_{22},\alpha_{i}} )} = {\mathbb{e}}^{{- 2}/T_{i}}},}\end{matrix}\quad} $ where k₁₁, k₁₂, k₂₁ and k₂₂ are vectors ofK_(i), α_(i) is an estimate index, r is the reference input signal,T_(i) is an amount of time for the resource management component toproduce the output signal comprising a steady state, and M_(i) is amaximum boundary for the output signal.
 12. The system of claim 8,wherein the controller gain is defined by the equation$\{ {\begin{matrix}{{f_{1}( {k_{1},k_{2}} )} = {{- 2}\;{\mathbb{e}}^{{- 4}/T_{i}}{\cos( {\pi\frac{\log\; r}{\log\; M_{i}}} )}}} \\{{{f_{2}( {k_{1},k_{2}} )} = {\mathbb{e}}^{{- 2}/T_{i}}},}\end{matrix}\quad} $ where k₁ and k₂ are vectors of K_(i), r isthe reference input signal, T_(i) is an amount of time for the resourcemanagement component to produce the output signal comprising a steadystate, and M_(i) is a maximum boundary for the output signal.
 13. Amethod for an electrical control system to manage in apoint-to-multipoint (P2MP) network that includes at least one rootstation and at least one leaf station that are each configured totransmit and receive one or more network packets over the P2MP network,the method for the electrical control system comprising: configuring acompensator for the electrical control system based, at least in part,on the equation ${F_{i} = {{\begin{bmatrix}K_{i} & 1\end{bmatrix}\begin{bmatrix}{A_{i} - I} & B_{i} \\C_{i} & 0\end{bmatrix}}^{- 1}\begin{bmatrix}0 \\1\end{bmatrix}}},{{where}\mspace{14mu}\begin{bmatrix}{A_{i} - I} & B_{i} \\C_{i} & 0\end{bmatrix}}$ is a non-singular matrix, where A_(i) is a state matrix,where B_(i) is an input matrix, where C_(i) is an output matrix, andwhere K_(i) is a controller gain for adjusting the output, wherein theelectrical control system includes a controller gain for an output, acompensator for adjusting a control error toward a zero value andadjusting the output toward a steady state; repeatedly analyzing theoutput of the electrical control system to determine if the output isequal to a desired output; dynamically adjusting the output of theelectrical control system to provide a new output by changing thecontroller gain until the new output achieves the desired output; andtransmitting one or more network packets to one or more of the at leastone leaf station and at least one root station based at least in part onthe output of the electrical control system; wherein the controller gainis K_(i), a constant matrix and is defined by the equation$\{ {\begin{matrix}{{f_{1}( {k_{11},k_{12},k_{21},k_{22},\alpha_{i}} )} = {{- 2}\;{\mathbb{e}}^{{- 4}/T_{i}}{\cos( {\pi\frac{\log\; r}{\log\; M_{i}}} )}}} \\{{{f_{2}( {k_{11},k_{12},k_{21},k_{22},\alpha_{i}} )} = {\mathbb{e}}^{{- 2}/T_{i}}},}\end{matrix}\quad} $ where k₁₁, k₁₂, k₂₁ and k₂₂ are vectors ofK_(i), α_(i) is an estimate index, r is a reference input, T_(i) is theamount of time for the electrical control system to produce an outputcomprising a steady state, and M_(i) is the maximum boundary for theoutput.
 14. The method of claim 13, wherein the electrical controlsystem implements a prediction-based resource allocation (PRA) scheme.15. The method of claim 13, wherein the electrical control system isbased, at least in part, on the equationx_(i)(n+1)=(A_(i)−B_(i)K_(i))x_(i)(n), where x_(i)(n) is a state vectorthat indicates a bandwidth requirement and queue length of the at leastone leaf station, where A_(i) is a state vector matrix, B_(i) is aninput vector matrix, where K_(i) is a constant matrix, and where n is agiven time.
 16. A method for an electrical control system to manageresources in a point-to-multipoint (P2MP) network that includes at leastone root station and at least one leaf station that are configured totransmit and receive one or more network packets over the P2MP network,the method for the electrical control system comprising: configuring acompensator for the electrical control system based, at least in part,on the equation ${F_{i} = {{\begin{bmatrix}K_{i} & 1\end{bmatrix}\begin{bmatrix}{A_{i} - I} & B_{i} \\C_{i} & 0\end{bmatrix}}^{- 1}\begin{bmatrix}0 \\1\end{bmatrix}}},{{where}\mspace{14mu}\begin{bmatrix}{A_{i} - I} & B_{i} \\C_{i} & 0\end{bmatrix}}$ is a non-singular matrix, where A_(i) is a state matrix,where B_(i) is an input matrix, where C_(i) is an output matrix, andwhere K_(i) is a controller gain for adjusting the output, wherein theelectrical control system includes a controller gain for an output, acompensator for adjusting a control error toward a zero value andadjusting the output toward a steady state; repeatedly analyzing theoutput of the electrical control system to determine if the output isequal to a desired output; dynamically adjusting the output of theelectrical control system to provide a new output by changing thecontroller gain until the new output achieves the desired output;transmitting one or more network packets to one or more of the at leastone leaf station and at least one root station based at least in part onthe output of the electrical control system; wherein the controller gainis K_(i), a constant matrix and is defined by the equation$\{ {\begin{matrix}{{f_{1}( {k_{1},k_{2}} )} = {{- 2}\;{\mathbb{e}}^{{- 4}/T_{i}}{\cos( {\pi\frac{\log\; r}{\log\; M_{i}}} )}}} \\{{{f_{2}( {k_{1},k_{2}} )} = {\mathbb{e}}^{{- 2}/T_{i}}},}\end{matrix}\quad} $ where k₁ and k₂ are vectors of K_(i), r is areference input, T_(i) is the amount of time for the electrical controlsystem to produce an output comprising a steady state, and M_(i) is themaximum boundary for the output.
 17. The method of claim 16, wherein theelectrical control system implements a request-based resource allocation(RRA) scheme.
 18. The method of claim 16, wherein the electrical controlsystem is based, at least in part, on the equationx_(i)(n+1)=(A_(i)−B_(i)K_(i))x_(i)(n), where x_(i)(n) is a state vectorthat indicates a bandwidth requirement and queue length of the at leastone leaf station, where A_(i) is a state vector matrix, B_(i) is aninput vector matrix, where K_(i) is a constant matrix, and where n is agiven time.
 19. A method for an electrical control system to manageresources in a point-to-multipoint (P2MP) network that includes at leastone root station and at least one leaf station each configured totransmit and receive one or more network packets over the P2MP network,the method for the electrical control system comprising: controlling theat least one root station based, at least in part, on output of theelectrical control system, wherein the electrical control system isconfigured to reduce an amount of time for the electrical control systemto produce an output comprising a steady state and also configured todefine a maximum boundary for the output, the electrical control systemcomprising a feedback control loop and a compensator, wherein thecompensator is configured to adjust a control error toward a zero valueand to adjust the output toward the steady state; transmitting the oneor more network packets to the at least one leaf station, wherein the atleast one root station is capable of transmitting and receiving the oneor more network packets from the at least one root station, and whereinthe at least one leaf station is capable of communication with the atleast one root station; and repeating the controlling and transmittingfor at least one of the one or more network packets; and wherein theelectrical control system is based, at least in part, on the equationx_(i)(n+1)=(A_(i)−B_(i)K_(i))x_(i)(n), where x_(i)(n) is a state vectorthat indicates a bandwidth requirement and queue length of the at leastone leaf station, where A_(i) is a state vector matrix, B_(i) is aninput vector matrix, where K_(i) is a constant matrix, and where n is agiven time.
 20. The method of claim 19, wherein each of the at least oneleaf station is prohibited from communicating with at least a portion ofthe other leaf stations.
 21. The method of claim 19, wherein thecontrolling further comprises: analyzing the output to determine if theoutput is equal to a desired output; if the output is not equal to thedesired output, adjusting the output by a controller gain to produce anew output; and repeating the analyzing and the adjusting until the newoutput is equal to the desired output.